"Naturally-occurring" or "programmed" cell death (PCD) in which the cell uses specialized cellular machinery to kill itself is a ubiquitous phenomenon that occurs early in organ development. Such a cell suicide mechanism that enables metazoans to control cell number and eliminate cells threatening the organism's survival has been thought to be under genetic control. In this article, we developed a novel statistical model for mapping specific genes or quantitative trait loci (QTL) that are responsible for the PCD process based on polymorphic molecular markers. This model incorporates the biological mechanisms of PCD that undergoes two different developmental stages, exponential growth and polynomial death. We derived a parametric approach to model the exponential growth and a nonparametric approach based on the Legendre function to model the polynomial death. A series of stationary and nonstationary models has been used to approximate the structure of the covariance matrix among cell numbers at a multitude of different times. The statistical behavior of our model is investigated through simulations studies and validated by a real example in rice.